The Comics
Arguably, Burton's movie didn't influence the comics directly as much as give them even more reason to pursue the dark, Frank Miller route they were already taking (Although 1992's "Destroyer" storyline recreated Gotham City using Anton Furst's production designs for the architecture of the movie, probably the most concrete example of the movie impacting the comic continuity outside of the temporary return of Vicki Vale for the first time in decades). One of the few things that the movie's success did was allow for DC to launch Legends of The Dark Knight, an anthology title that was also the first new ongoing solo Batman monthly series in 49 years; another was the much-told (and possibly apocryphal) story about Warners demanding that Grant Morrison and Dave McKean's Arkham Asylum: A Serious House On Serious Earth graphic novel was stripped of some of its more risque Joker scenes before release (Sorry, those who wanted to see the Joker dressed as Madonna and pinching Batman's ass). Otherwise, the comics kept on doing what they were already doing.

(There's arguably a case to be made for the idea that Batman's success drove new readers and, perhaps more importantly, movie and television producers to the medium, leading to the early 1990s speculator boom and subsequent bust, but that's another article in itself.)

The Movies
If there's one thing Hollywood likes more than a hit, it's a formula for more hits, and Batman's success convinced movie executives that they could do exactly the same thing with whatever comic character they wanted (Even if, in Robert Townsend's case, he had to make him up). The result? Lots of bad movies, made with less love and less talent than Burton brought to Batman. Exhibit A: Warren Beatty's garish, flat Dick Tracy:

Looking at some of these, I'm kind of glad that Batman and Robin accidentally killed the genre for a few years.

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The Best Thing To Have Come From The Success Of The Movie
Surely there's no contest, right...?

Batman: The Animated Series (AKA The Adventures of Batman & Robin and The New Batman Adventures, amongst other names it had during its brief but wonderful life) may have gained from the success of Tim Burton's movies - it premiered in 1992, following the release of Batman Returns, but had been in the works long before - but it wasn't a copy by any stretch of the imagination. Dark without being humorless, endlessly stylish thanks to the talents of Bruce Timm, Dan Riba and many others and much smarter than other Saturday morning cartoons (or, for that matter, many other incarnations of Batman), Batman: The Animated Series defined the character for a generation, and remains an example of how good an animated series can be.

Maybe you'd like to see the original animation that got the series greenlit...?